Finite Group Databasebeta

← #411950

Subgroups — #411950

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2C2×#2024C212yesmaximal
3C2×#1803C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6#11384C212yesmaximal
7#11293C212yesmaximal
8#11067C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2551—24no—
21#2551—24no—
22C2×#395—22no—
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2024—28no—
31#1803—24no—
32C4×#103—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1441—22no—
43C4⋊#95—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C23×Q8—22no—
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—28no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—44no—
59C2×C4⋊Q8—44no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8Q814yescomplemented
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103D414yes—
95C2×#103Q814yescomplemented
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99#395—216no—
100#391—24no—
101#391—42no—
102#389—28no—
103#389—24no—
104#389—24no—
105#384—22no—
106#384—22no—
107#384—22no—
108#384D414yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4D414yes—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D412yes—
120#340—24no—
121#340—24no—
122#340—22no—
123#340—22no—
124#340—24no—
125#340—24no—
126#340—42no—
127#336—24no—
128#336—42no—
129C22×Q8—28no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—44no—
142C22×Q8—44no—
143C22×Q8—44no—
144C22×Q8—44no—
145C22×Q8—82no—
146C22×Q8—82no—
147C22×Q8C2×D412yes—
148C22×Q8C2×Q812yes—
149C22×Q8C2×Q812yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8C2×D412yes—
181C22×C8C2×Q812yes—
182C4⋊Q8—48no—
183C4⋊Q8—48no—
184C4⋊Q8—48no—
185C4⋊Q8—432no—
186C4⋊Q8—48no—
187C4×Q8—28no—
188C4×Q8—28no—
189C4×Q8—44no—
190C4×Q8—44no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—81no—
235C2×C4⋊C4—81no—
236C2×C4⋊C4—81no—
237C2×C4⋊C4—81no—
238C2×C4⋊C4SD1614yes—
239C2×C4⋊C4Q1614yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×Q812yes—
245C2×C4⋊C4C2×Q812yes—
246C2×C4⋊C4#4112yes—
247C2×C4⋊C4#4112yes—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×Q812yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266C4⋊C8—44no—
267#103—28no—
268#103—28no—
269#103—48no—
270C4×C8—44no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—44no—
275#95—44no—
276#95—82no—
277#95—82no—
278C24C22×D411yescharacteristic
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—416no—
295C2×Q8—416no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—88no—
309C2×Q8—88no—
310C2×Q8—88no—
311C2×Q8—88no—
312C2×Q8—88no—
313C2×Q8C4⋊Q814yescomplemented
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4#12012yes—
373C22×C4#12012yes—
374C22×C4C22⋊Q812yes—
375C22×C4C22⋊Q812yes—
376C22×C4#12412yes—
377C22×C4#12412yes—
378C22×C4C2×SD1614yes—
379C22×C4C2×SD1612yes—
380C22×C4C2×Q1614yes—
381C22×C4C2×Q1612yes—
382C22×C4AGL(1,ℤ/8)12yes—
383C22×C4#13712yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×Q811yescharacteristic
388C22×C4C22×Q811yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C42−1+411yescharacteristic
397C22×C42−1+411yescharacteristic
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—416no—
432C4⋊C4—416no—
433C4⋊C4—42no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4#12114yes—
447C4⋊C4C22⋊Q814yes—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—48no—
453C42—48no—
454C42C22⋊Q814yes—
455C42C22⋊Q814yes—
456C42#12414yes—
457C42#12414yes—
458C23C2×#12011yescharacteristic
459C23C2×C22⋊Q811yescharacteristic
460C23C2×#12411yescharacteristic
461C23D4211yescharacteristic
462C23#54611yescharacteristic
463C23#54811yescharacteristic
464C23D4×Q811yescharacteristic
465C23C22×SD1611yescharacteristic
466C23C22×Q1611yescharacteristic
467C23C2×AGL(1,ℤ/8)11yescharacteristic
468C23C2×#13711yescharacteristic
469C23#57611yescharacteristic
470C23#57711yescharacteristic
471C23#57711yescharacteristic
472C23#57811yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—832no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#39314yes—
545C2×C4#39314yes—
546C2×C4#45014yes—
547C2×C4#45114yes—
548C2×C4#47514yes—
549C2×C4#47714yes—
550C2×C4#48714yes—
551C2×C4#48914yes—
552C2×C4C2×#12111yescharacteristic
553C2×C4C2×C22⋊Q811yescharacteristic
554C2×C4C2×C22⋊Q811yescharacteristic
555C2×C4C2×C22⋊Q811yescharacteristic
556C2×C4C2×#12411yescharacteristic
557C2×C4C2×#12411yescharacteristic
558C2×C4C2×C4⋊Q811yescharacteristic
559C2×C4#53211yescharacteristic
560C2×C4#53311yescharacteristic
561C2×C4#53311yescharacteristic
562C2×C4#53511yescharacteristic
563C2×C4#53611yescharacteristic
564C2×C4#54211yescharacteristic
565C2×C4#54211yescharacteristic
566C2×C4#54411yescharacteristic
567C2×C4#54411yescharacteristic
568C2×C4D4211yescharacteristic
569C2×C4#54611yescharacteristic
570C2×C4#54711yescharacteristic
571C2×C4#54711yescharacteristic
572C2×C4#54811yescharacteristic
573C2×C4D4×Q811yescharacteristic
574C2×C4#55411yescharacteristic
575C2×C4#55411yescharacteristic
576C2×C4#55511yescharacteristic
577C2×C4#55511yescharacteristic
578C2×C4#55611yescharacteristic
579C2×C4#55611yescharacteristic
580C8—42no—
581C8—42no—
582C8—82no—
583C22C2×#39311yescharacteristic
584C22#240811yescharacteristic
585C22#240911yescharacteristic
586C22#243911yescharacteristic
587C22#244011yescharacteristic
588C22#249111yescharacteristic
589C22#249211yescharacteristic
590C22C2×#45011yescharacteristic
591C22C2×#45111yescharacteristic
592C22#301211yescharacteristic
593C22#301311yescharacteristic
594C22C2×#47511yescharacteristic
595C22C2×#47711yescharacteristic
596C22#308311yescharacteristic
597C22#308411yescharacteristic
598C22C2×#48711yescharacteristic
599C22C2×#48911yescharacteristic
600C22#313611yescharacteristic
601C22#313711yescharacteristic
602C22#327411yescharacteristic
603C22#327811yescharacteristic
604C22D4×SD1611yescharacteristic
605C22D4×Q1611yescharacteristic
606C22#329511yescharacteristic
607C22#329911yescharacteristic
608C22#332411yescharacteristic
609C22#332811yescharacteristic
610C22#335911yescharacteristic
611C22#336311yescharacteristic
612C22#337311yescharacteristic
613C22#337511yescharacteristic
614C22Q8×SD1611yescharacteristic
615C22Q8×Q1611yescharacteristic
616C22#338611yescharacteristic
617C22#338711yescharacteristic
618C4—24no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—44no—
624C4—82no—
625C4—84no—
626C4—162no—
627C4#202418yescomplemented
628C4#243911yescharacteristic
629C4#244011yescharacteristic
630C4#249111yescharacteristic
631C4#249211yescharacteristic
632C4#254711yescharacteristic
633C4#256311yescharacteristic
634C4#262811yescharacteristic
635C4#285711yescharacteristic
636C2#1512511yescharacteristic
637C2C2×#202411yescharacteristic
638C2#2345811yescharacteristic
639C2#2424011yescharacteristic
640C2#2424311yescharacteristic
641C2#2433011yescharacteristic
642C2#2433311yescharacteristic
643C2#2602211yescharacteristic
644C2#2602611yescharacteristic
645C2#2604211yescharacteristic
646C2#2604611yescharacteristic
647C2#2776911yescharacteristic
648C2#2777811yescharacteristic
649C2#2781511yescharacteristic
650C2#2782311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
65. C2×C4×Q8Q8semidirect
95. C2×#103Q8semidirect
313. C2×Q8C4⋊Q8semidirect
627. C4#2024semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#25512
C2×#20242
C2×#18032
C2×C4×#1032
#139262
#113842
#112932
#110672
#108492
#108412
#108342